Due to the discrete \(\mathbb {Z}_2\)  symmetry, the scalar multiplet dark matter model exhibits an interesting feature: multi-step phase transition could happen due to a multi-degenerate minimum. Usually the multi-step happens due to parameter choice, there is no symmetry to guarantee it happens [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}, [34]}, [35]}, [36]}, [37]}, [38]}, [39]}, [40]}, [41]}, [42]}, [43]}, [44]}, [45]}, [46]}, [47]}, [48]}, [49]}, [50]}, [51]}, [52]}, [53]}, [54]}, [55]}, [56]}, [57]}, [58]}, [59]}, [60]}, [61]}, [62]}, [63]}, [64]}, [65]}, [66]}, [67]}, [68]}, [69]}, [70]}, [71]}, [72]}, [73]}, [74]}, [75]}, [76]}, [77]}, [78]}, [79]}, [80]}, [81]}, [82]}, [83]}, [84]}, [85]}, [86]}, [87]}, [88]}, [89]}, [90]}, [91]}, [92]}, [93]}, [94]}, [95]}, [96]}, [97]}, [98]}, [99]}, [100]}, [101]}, [102]}, [103]}, [104]}, [105]}, [106]}, [107]}, [108]}, [109]}, [110]}, [111]}, [112]}, [113]}, [114]}, [115]}, [116]}, [117]}, [118]}, [119]}, [120]}, [121]}, [122]}, [123]}, [124]}, [125]}, [126]}, [127]}, [128]}, [129]}, [130]}, [131]}, [132]}, [133]}, [134]}, [135]}, [136]}, [137]}, [138]}, [139]}, [140]}, [141]}, [142]}, [143]}, [144]}, [145]}, [146]}, [147]}, [148]}, [149]}, [150]}, [151]}, [152]}, [153]}, [154]}, [155]}, [156]}, [157]}, [158]}, [159]}, [160]}, [161]}, [162]}, [163]}, [164]}, [165]}. On the other hand, a discrete symmetry could guarantee the potential form with a multi-degenerate vacuum. This provides room for a multi-step phase transition to happen. More interestingly, the GW spectra could have one or multi peaks through the multi-step phase transition if several phase transitions in the multi-steps are first order. Therefore, it is possible to use the observation of the multi-peak GW spectra to probe the new physics models with the multi-step transition.
